Three points of a triangle are among 8 vertex of a cube. So the number of such acute triangles is $\text{(A)}0\qquad\text{(B)}6\qquad\text{(C)}8\qquad\text{(D)}24$

Prove that for any tetrahedron the radius of the inscribed sphere $r <\frac{ ab}{ 2(a + b)}$, where $a$ and $b$ are the lengths of any pair of opposite edges.

$SABCD$ is quadrangular pyramid. Lateral faces are acute triangles with orthocenters lying in one plane. $ABCD$ is base of pyramid and $AC$ and $BD$ intersects at $P$, where $SP$ is height of pyramid. Prove that $AC \perp BD$

A plastic snap-together cube has a protruding snap on one side and receptacle holes on the other five sides as shown. What is the smallest number of these cubes that can be snapped together so that only receptacle holes are showing? [asy] draw((0,0)--(4,0)--(4,4)--(0,4)--cycle); draw(circle((2,2),1)); draw((4,0)--(6,1)--(6,5)--(4,4)); draw((6,5)--(2,5)--(0,4)); draw(ellipse((5,2.5),0.5,1)); fill(ellipse((3,4.5),1,0.25),black); fill((2,4.5)--(2,5.25)--(4,5.25)--(4,4.5)--cycle,black); fill(ellipse((3,5.25),1,0.25),black); [/asy] $\text{(A)}\ 3 \qquad \text{(B)}\ 4 \qquad \text{(C)}\ 5 \qquad \text{(D)}\ 6 \qquad \text{(E)}\ 8$

Height of a circular truncated cone is $8$. Center of sphere $O_1$ with a radius of $2$ is on the axis of the circular truncated cone. Sphere $O_1$ is tangent to the top surface and the flank. We can put another sphere $O_2$, satisfying that sphere $O_2$ with a radius of $3$ have only one common point with sphere $O_1$, bottom surface and the flank. Besides $O_2$, how many spheres can we put inside the circular truncated cone? $\text{(A)}1\qquad\text{(B)}2\qquad\text{(C)}3\qquad\text{(D)}4$

The midpoint of the edge $SA$ of the triangular pyramid of $SABC$ has equal distances from all the vertices of the pyramid. Let $SH$ be the height of the pyramid. Prove that $BA^2 + BH^2 = C A^2 + CH^2$.

Given a rectangular strip of measure $12 \times 1$. Paste this strip in two layers over the cube with edge $1$ (the strip can be bent, but cannot be cut). (V. Shevyakov)

In each lateral face of a pentagonal prism at least one of the four angles is equal to $f$. Find all possible values of $f$. (A Shapovalov)

Prove the theorem: In a tetrahedron, the plane bisector of any dihedral angle divides the opposite edge into segments proportional to the areas of the tetrahedron faces that form this dihedral angle.

Consider a right circular cylinder with radius $ r$ of the base, hight $ h$. Find the volume of the solid by revolving the cylinder about a diameter of the base.

Actually, this is an MO I participated in :) but it's really hard to get problems from this year if you don't know some people. P1. Let $f$ be a function satisfying $f(x + 1) + f(x - 1) = \sqrt{2} f(x)$, for all reals $x$. If $f(x - 1) = a$ and $f(x) = b$, determine the value of $f(x + 4)$. [hide=Remarks]We found out that this is the modified version of a problem from LMNAS UGM 2008, Senior High School Level, on its First Round. This is also the same with Arthur Engel's "Problem Solving Strategies" Book, Example Problem E2.[/hide] P2. The sequence of "Sanga" numbers is formed by the following procedure. i. Pick a positive integer $n$. ii. The first term of the sequence $(U_1)$ is $9n$. iii. For $k \geq 2$, $U_k = U_{k-1} - 17$. Sanga$[r]$ is the "Sanga" sequence whose smallest positive term is $r$. As an example, for $n = 3$, the "Sanga" sequence which is formed is $27, 10, -7, -24, -41, \ldots.$ Since the smallest positive term of such sequence is $10$, for $n = 3$, the sequence formed is called Sanga$[10]$. For $n \leq 100$, determine the sum of all $n$ which makes the sequence Sanga$[4]$. P3. The cube $ABCD.EFGH$ has an edge length of 6 cm. Point $R$ is on the extension of line (segment) $EH$ with $EH : ER = 1 : 2$, such that triangle $AFR$ cuts edge $GH$ at point $P$ and cuts edge $DH$ at $Q$. Determine the area of the region bounded by the quadrilateral $AFPQ$. [url=https://artofproblemsolving.com/community/q1h2395046p19649729]P4[/url]. Ten skydivers are planning to form a circle formation when they are in the air by holding hands with both adjacent skydivers. If each person has 2 choices for the colour of his/her uniform to be worn, that is, red or white, determine the number of different colour formations that can be constructed. P5. After pressing the start button, a game machine works according to the following procedure. i. It picks 7 numbers randomly from 1 to 9 (these numbers are integers, not stated but corrected) without showing it on screen. ii. It shows the product of the seven chosen numbes on screen. iii. It shows a calculator menu (it does not function as a calculator) on screen and asks the player whether the sum of the seven chosen numbers is odd or even. iv. Shows the seven chosen numbers and their sum and products. v. Releases a prize if the guess of the player was correct or shows the message "Try again" on screen if the guess by the player was incorrect. (Although the player is not allowed to guess with those numbers, and the machine's procedures are started all over again.) Kiki says that this game is really easy since the probability of winning is greater than $90$%. Explain, whether you agree with Kiki.

Consider a piece of material in the shape of a right circular conical frustum with radii $R,r,R>r$. A cavity in the shape of another coaxial right circular conical frustum was drilled into the material (see the picture). That way only half of the original volume of material remained. Compute radii $R',r'$ of the cavity. Decide for which ratio $R/r$ the problem has a solution. [img]https://cdn.artofproblemsolving.com/attachments/b/f/12f579458b7cf0fc31849b319e6f58e50b0363.png[/img]

A closed container in the shape of a rectangular parallelepiped contains $1$ liter of water. If the container rests horizontally on three different sides, the water level is $2$ cm, $4$ cm and $5$ cm. Calculate the volume of the parallelepiped.

A cylinder with radius $5$ and height $1$ is rolling on the (unslanted) floor. Inside the cylinder, there is water that has constant height $\frac{15}{2}$ as the cylinder rolls on the floor. What is the volume of the water?

A pyramid has a square base with sides of length 1 and has lateral faces that are equilateral triangles. A cube is placed within the pyramid so that one face is on the base of the pyramid and its opposite face has all its edges on the lateral faces of the pyramid. What is the volume of this cube? $ \textbf{(A)}\ 5\sqrt{2}-7 \qquad \textbf{(B)}\ 7-4\sqrt{3} \qquad \textbf{(C)}\ \frac{2\sqrt{2}}{27} \qquad \textbf{(D)}\ \frac{\sqrt{2}}{9} \qquad \textbf{(E)}\ \frac{\sqrt{3}}{9} $

Let $V$ be a rectangular prism with integer side lengths. The largest face has area $240$ and the smallest face has area $48$. A third face has area $x$, where $x$ is not equal to $48$ or $240$. What is the sum of all possible values of $x$?

In the tetrahedron $ABCD$ three of the faces are right-angled triangles and the other is not an obtuse triangle. Prove that: (a) the fourth wall of the tetrahedron is a right-angled triangle if and only if exactly two of the plane angles having common vertex with the some of vertices of the tetrahedron are equal. (b) its volume is equal to $\frac16$ multiplied by the multiple of two shortest edges and an edge not lying on the same wall.

Find the minimal $k$ such that every set of $k$ different lines in $\mathbb R^3$ contains either $3$ mutually parallel lines or $3$ mutually intersecting lines or $3$ mutually skew lines.

Find $n$ such that $n - 76$ and $n + 76$ are both cubes of positive integers.

Let $PA_1A_2...A_{12} $ be the regular pyramid, $ A_1A_2...A_{12} $ is regular polygon, $S$ is area of the triangle $PA_1A_5$ and angle between of the planes $A_1A_2...A_{12} $ and $ PA_1A_5 $ is equal to $ \alpha $. Find the volume of the pyramid.

Let $SABC$ be a regular pyramid, $O$ the center of basis $ABC$, and $M$ the midpoint of $[BC]$. If $N \in [SA]$ such that $SA = 25 \cdot NS$ and $SO \cap MN=\{P\}$, $AM=2\cdot SO$, prove that the planes $(ABP)$ and $(SBC)$ are perpendicular.

Given a cube. On each edge of the cube, we write a number, either $1$ or $-1$. For each face of the cube, we multiply the four numbers on the edges of this face, and write the product on this face. Finally, we add all the eighteen numbers that we wrote down on the edges and face of the cube. What is the smallest possible sum that we can get?

The faces of a box with integer edge lengths are painted green. The box is partitioned into unit cubes. Find the dimensions of the box if the number of unit cubes with no green face is one third of the total number of cubes.

A set $ S$ of points from the space will be called [b]completely symmetric[/b] if it has at least three elements and fulfills the condition that for every two distinct points $ A$ and $ B$ from $ S$, the perpendicular bisector plane of the segment $ AB$ is a plane of symmetry for $ S$. Prove that if a completely symmetric set is finite, then it consists of the vertices of either a regular polygon, or a regular tetrahedron or a regular octahedron.

In the $xyz$ space with the origin $O$, Let $K_1$ be the surface and inner part of the sphere centered on the point $(1,\ 0,\ 0)$ with radius 2 and let $K_2$ be the surface and inner part of the sphere centered on the point $(-1,\ 0,\ 0)$ with radius 2. For three points $P,\ Q,\ R$ in the space, consider points $X,\ Y$ defined by \[\overrightarrow{OX}=\overrightarrow{OP}+\overrightarrow{OQ},\ \overrightarrow{OY}=\frac 13(\overrightarrow{OP}+\overrightarrow{OQ}+\overrightarrow{OR}).\] (1) When $P,\ Q$ move every cranny in $K_1,\ K_2$ respectively, find the volume of the solid generated by the whole points of the point $X$. (2) Find the volume of the solid generated by the whole points of the point $R$ for which for any $P$ belonging to $K_1$ and any $Q$ belonging to $K_2$, $Y$ belongs to $K_1$. (3) Find the volume of the solid generated by the whole points of the point $R$ for which for any $P$ belonging to $K_1$ and any $Q$ belonging to $K_2$, $Y$ belongs to $K_1\cup K_2$.